I need to show the $\mathcal Z$-transform of $y[n]$ as a function of the $\mathcal Z$-transform of $x[n]$.
Now I know that for downsampling alone:
$$Y(z) = \frac1M\sum_{m=0}^{M-1} X\left(e^\frac{-j2\pi{m}}M\cdot z^\frac 1M\right)$$
And that for upsampling alone:
$$Y(z) = X\left(z^M\right)$$
My problem here is that I don't quite understand how upsampling would affect the downsampling equation. Say we call the point between downsampling and upsampling $w[n]$, we get:
\begin{align}Y(z) &= W\left(z^M\right)\\ W(z) &= \frac1M\sum_{m=0}^{M-1} X\left(e^\frac{-j2\pi{m}}M\cdot z^\frac 1M\right)\end{align}
I'm not sure if I will be getting:
$$Y(z) = \frac1M\sum_{m=0}^{M-1} X\left(e^\frac{-j2\pi{m}}M\cdot z\right)$$
Or:
$$Y(z) = \frac1M\sum_{m=0}^{M-1} X\left(e^{-j2\pi{m}}\cdot z\right)$$
Or even something else.
I'll be very happy for some clarification on this subject because I'm very baffled by it and I cannot find a solution anywhere, thank you.